The Inflation Forecast Competition;
Different Approaches to Forecast Dutch Inflation
Master Thesis
Wouter Bastiaan van Aanholt∗ 1872869
Supervisor: dr. Diego Ronchetti
January 28, 2016
Abstract
This paper investigates the predictive powers of different versions of the Phillips curve and the BVAR to forecast inflation for a one-year ahead forecast and a two-year ahead forecast, reported on both a quarterly basis as a yearly basis. We benchmark these models against the CPB model SAFFIER. The results show that the predic-tive powers of the BVAR is statistically similar to the benchmark model SAFFIER. Furthermore found that different versions of the Phillips curve are significantly out-perform by the BVAR as the benchmark SAFFIER. Only for the one-year ahead forecast on a quarterly yearly basis the different versions of the Phillips curve are not significantly outperformed by the BVAR.
Keywords: Inflation, Forecast, CPB, Phillips curve, New Keynesian Phillips curve, BVAR
∗
1
Introduction
Inflation, a general rising of the price level of goods and services, is important for households, investors and policymakers. Therefore, forecasting inflation is a major area in economic research. This paper compares different economic approaches to forecast inflation, both on predictive accuracy and on economic theory. One of these approaches is the cost based approach. This approach is used by several institutions, including but not limited to the Netherlands Bureau for Economic Policy Analysis (CPB) and the Dutch National Bank (DNB).
The CPB forecasts the whole Dutch economy with their Short and medium term Analysis and Forecasting using Formal Implementation of Economic Reasoning (SAFFIER) model, including inflation. In SAFFIER, inflation is related to the cost develop-ments, and the most important elements are wage developdevelop-ments, imported goods and services prices, rent prices, energy prices and health care. A disadvantage of the SAFFIER model is that the influence of monetary variables is not included. It is also uncertain how fast the cost developments are passed on to prices and whether this pass on is constant over time. The SAFFIER forecasts, made by the CPB, are used as an exogenous benchmark.
In this study, the predictive quality of different inflation forecast approaches is as-sessed. These approaches are chosen in such a way that they include the most relevant views on inflation. In this way, it is possible to evaluate whether differ-ent approaches are better than the currdiffer-ently used model, SAFFIER. Therefore, this research will evaluate each approach on both the predictive accuracy, by means of a forecast competition, and on the economic foundation on which it is built, by a literature review.
2
Literature research
In the literature on inflation, for the long term, the quantity theory of money is generally accepted. The formula of the theory is given by
M V = P Q,
where M, V, P and Q denote the nominal quantity of money (M), the velocity of money in final expenditures (V), the general price level (P), and an index of the real value of final expenditures (Q), respectively.
This paper is focused on forecasting on the short term. The inflation literature shows many different views. Keynes proposed an useful theory for the short term. The Keynesian view entails that money supply has a more indirect effect on in-flation. It is caused, according to Keynes, by three points: demand outstripping supply, inflation built into the system, and shocks in costs. Since this research will focus on forecasting inflation in the short term, the Keynesian view provides a good framework for the causes of short term inflation.
2.1 Phillips curve
In Phillips (1958), the relationship between inflation and unemployment was found. This, together with the Keynesian economic model, creates an useful framework for economic policy. In this paper, we interpret a Phillips curve more broadly and similar to Stock and Watson (2008), including all forecasts that use one or multiple activity variables to forecast inflation or the change in inflation, where activity variables show a correlation with the activity of the economy. In the traditional Phillips curve the activity variable is the unemployment rate. Other examples of activity variables are output gap, output growth or utilization rate of industry.
Gordon (1990) introduced a different form of the Phillips curve, the ”triangle model”, which does not only include lagged inflation and activity variables, but also variables that can indicate supply shocks. In Stock and Watson (2008) the triangle model per-forms episodically really well, particularly during the high unemployment disinflation of the early 1980s and also in the late 1990s on U.S. data. Note that the ”triangle model” has many similarities to the Keynesian view. All causes of inflation Keynes describes are represented in this model.
2.2 New Keynesian Phillips curve
The Phillips curve is originally macro founded, historical patterns are described in formulas. Lucas (1976) notes that these relations are not based on policy-invariant variables. Therefore the relations are bound to change if policies change. Hence the outcome of policy changes based on these model are misleading.
The Lucas Critique inspired new fields of research. One of which is the New Key-nesian model approach. This approach combines the dynamic stochastic general equilibrium models of the Real Business Cycle theory with the assumptions of the classical monetary models. The key elements of the New Keynesian approach are monopolistic competition, nominal rigidities and short run non-neutrality of mone-tary policy.
One of the key building blocks of the New Keynesian Model Approach is the New Keynesian Phillips curve (NKPC). The NKPC is constructed based on the assump-tions mentioned above.
The basic NKPC is given as
πt= πet + αyt,
where πtis the rate of inflation at time t, ytdenotes the output gap at time t and πte
The fundamental distinction of the NKPC from the Phillips curve is the input of inflation in the formulation of πte. Gal´ı and Gertler (1999) propose that πet is given by
πte= αfEt(πt+1) + αbπt−1
where Et(πt+1) and πt−1 denote the expected inflation at time t + 1 with the
infor-mation available at time t.
The Phillips curve has the implicit assumption that αf is zero, the NKPC imposes
the convex restriction αf + αb = 1, and 0 ≤ αf, αb ≤ 1.
If we assume αf = 0 and αb = 1 we have a Phillips curve with, dominantly,
in-flation stickiness. In the case we assume αf = 1 and αb = 0 we have a pure forward
looking NKPC with price stickiness. This indicates that inflation is only depending on the expected discount sum of the current and future value of the output gap.
Multiple papers show that there is indeed an important role for forward looking inflation Gal´ı and Gertler (1999), Zhang et al. (2008), Sbordone (2002), Gal´ı et al. (2002). On the other hand there are also studies that show the opposite Fuhrer and Moore (1995), Fuhrer (1997), Rudebusch (2002) and Roberts (2005). The main difference between these papers relates to the driver of inflation; the forward look-ing advocates argue that inflation is only dependlook-ing on the expected discount sum of the current and future value of the output gap or other activity variables, while backwards looking advocates argue that lagged inflation also drives inflation.
2.3 Taylor rule
The Taylor rule has multiple forms; we discussed the most general one, given by it= rt∗+ π∗+ β(πt− π∗) + γ(yt− yN) + t,
where it, r∗, πt, π∗, yt, yN and t denote the marginal lending rate, long-run real
in-terest rate, inflation rate, target inflation rate, GDP and real trend GDP or potential GDP, and the error term, respectively. In Taylor (1993) it is argued that π∗ and r∗ are both equal to 2 and β and γ are 1.5 and 0.5, respectively.
The Taylor rule is commonly used, however it has not been used to forecast infla-tion. The Taylor rule can also be seen as a Phillips curve with a monetary variable, by rewriting the Taylor rule as inflation being regressed on the output gap and the marginal lending rate, which is in principle a Phillips curve.
The Taylor rule is heavily discussed. Most of these discussion are not relevant for our study since we do not use the Taylor rule as a policy measure, but as a framework for a forecasting tool. Svensson (2003), however, makes a point that does influence our discussion. Svensson (2003) argues that for smaller and more open economies than the USA, economic state variables are important. State variables can be defined as variables that display the dynamics between countries, i.e. real exchange rate, terms of trade, foreign output and foreign interest rate. Since the Netherlands is a rela-tively small and open economy this can be relevant. McCallum and Nelson (2005) counter this argument by stating that there are no clear quantitative nor qualita-tive reasons that other state variables need to be included. Clarida et al. (2005) support this claim, since they did not find the need for an extra state variable, be-sides inflation and output gap, in their welfare function for the open economy model.
2.4 Bayesian VAR
Sims (1980) proposed a different view to model inflation, the vector autoregession (VAR). He criticises macroeconomics based on business cycles, fluctuation in ag-gregate measures of economic activity and fluctuation in prices. Both the Real Business Circle theory and the New Keynesian model approach are based on these phenomenons. The most profound critique relates to how these theories are iden-tified. The ”one-equation-at-a-time” specification that these models use, give the only allusion of identification. Hence the structure seems to solve a problem that it does not solve. Since a VAR system is providing a statistical method to estimate economic relationships, it is not affected by the same problems. Furthermore Sims (1980) advocates the uses of the Bayesian methods or other shrinking devices to improve the outcomes.
Techniques of forecasting using vector autoregession (VAR) were suggested in Litter-man (1976). This paper used Bayesian mathematics to develop these techniques. Lit-terman’s study resulted in the Litterman prior or the Minnesota prior of the BVAR. The main problem with a VAR is over-identification. VAR’s are usually based on a small number of variables. Litterman (1986) found that applying Bayesian shrinkage in the VAR containing only six variables can result into better forecasting perfor-mance. The Bayesian shrinking gives the freedom to incorporate far more variables. In many studies the Litterman prior has been a standard tool for applied macroeco-nomics, for example in Leeper et al. (1996), Sims and Zha (1998), and Robertson and Tallman (1999). It is shown by Ba´nbura et al. (2010), that large models’ forecasts outperform the forecasts of smaller models, but this result is already obtained by a system of 20 key macroeconomic indicators. Hence, there is no need for models with many variables.
2.5 SAFFIER
The SAFFIER model, as a cost based approach, assumes that prices are set based on the cost of the products plus a profit margin. For homogeneous products this is a realistic approach, but for more heterogeneous products it could be argued other-wise. For heterogeneous goods the price is set based on added value of the product or service to the costumer, or the willingness to pay for a product or service. This will not change if the cost of producing changes. The newer the industry, the more we see value based prices, which makes the cost based approach less predictive. In the long term it is very hard for a product or service to stay unique and therefore heterogeneous. So in the long term this model would hold.
Therefore the cost based approach only takes implicit notion of the supply demand dynamics due to the assumption that cost changes is incorporated in the prices. This is only partly true in the short term. Another shortcoming of the model is that it neglects the monetary side of inflation. This can have an influence on the willingness to invest of companies or people and is therefore important.
Nevertheless, the model is very specific and forecasts all the building blocks of infla-tion separately. Furthermore SAFFIER is build to incorporate policy changes. Also the outcomes of SAFFIER are influenced by an expert opinion. For those reasons, even though the economic foundation is not optimal, SAFFIER provides very good results.
3
Data
The data set consists of quarterly data on a quarter on quarter basis. This research uses thirteen time series and are all seasonally adjusted. Seven of these time series have data from 1980Q1 - 2014Q4, which give 140 data points. Five of the time series have data from 1988Q1 - 2014Q4, which result into 112 data points. One time series, the expected inflation, has only data from 1999Q1 - 2014Q4, which results into 60 data points. Furthermore the data is recorded in quarter to quarter format. Most data for this paper originates from government organisations CBS and DNB. Some of these data are also used for the economic forecast on the development of the Dutch economy of the CPB. In case quarterly data was not available, but a higher frequency was, the average value over the quarter is taken.
The data is given on a quarter on quarter basis. Hence, so are the forecasts. Most data is communicated in a year to year format over one whole year. The same holds for the SAFFIER forecast, therefore our quarterly data on quarter on quarter basis must first be formatted into quarterly data on year on year basis. This can then be formatted to yearly data on a year on year basis.
3.1 Inflation
Currently, inflation is measured by the percentage change in prices of certain goods and services. The goods and services are chosen by Dutch Statistics for the Nether-lands or by the ECB for Europe. In this research the consumer price index, CPI, is used to measure inflation. In figure 1 the historical developments of inflation are shown.
Figure 1: Historical developments of Inflation
3.2 Output gap
The output gap is the difference between the actual gross domestic product (GDP) and the potential GDP. The output Gap is in general set in percentages of the potential GDP. Hence the output gap is given by
OutGapt=
Yt− Yt∗
Yt∗ ,
where Ytand Yt∗ are the Actual GDP and potential GDP at time t, respectively.
The measurement of the potential GDP can be done in many different ways. The most commonly used method is the Hodrick-Prescott (HP) filter. The filter was in-troduced in a study of business cycles in 1980 of Hodrick and Prescott. The filter has become increasingly popular among econometricians. In 1997 Hodrick and Prescott finally published their paper Hodrick and Prescott (1997). The filter optimizes the signal for trend in the smooth trend model based on the mean squared error. In this paper we use the smoothness factor of 1600, since that is the default option for quarterly data. Other smoothness factors around 1600 provide an output gap with a similar pattern and only negligible differences. Hence there is no need to use another smoothness factor. In Figure 2 the potential GDP and the actual GDP are shown.
4
Methodology
In this section an outline of the set up of this research is given. First, we dive into the competition between the five different approaches. Then we look at the models that compete in our forecast competition.
4.1 Forecast competition setup
The five different approaches provide eight year of quarterly forecasts for inflation. The forecasts competition is conducted over a period of eight years from 2007 till 2014 for the quarterly forecast and over a period of seven years from 2008 until 2014 for yearly forecasts. The reason for this is that our data is on a quarter on quarter format. To get to a yearly forecast of inflation, we need to transform the data to year on year format. This is at a cost one-year of forecast information. Note that yearly and the quarterly forecasts are based on the same data and the same forecast, only the horizon of the forecasts shifts. The main reason for the small sample is the lack of data of the variable expected inflation. Furthermore this forecast competition is done based on two time lines. A one-year ahead forecast, for which we use the data till the last quarter of the previous year. So for example to forecast 2012Q1, 2012Q2, 2012Q3, and 2012Q4, the data till 2011Q4 is used. For a two-year ahead forecast the data until the last quarter of the year before the previous year is used. So for ex-ample to forecast 2012Q1, 2012Q2, 2012Q3, and 2012Q4, the data till 2010Q4 is used.
The models are compared with multiple statistics. The mean absolute forecast error (MAFE) and on the root mean square forecast error (RMSFE) are used to measure the accuracy of the forecast. They are given by
MAFE = P ∀t|At− Ft| n , and RMSFE = r P ∀t(At− Ft)2 n ,
where At is the actual level of inflation at time t, Ft is the forecast level of inflation
at time t and n is the number of forecasts in the sample.
5
Inflation models
In this section we discuss the specifications of the different models and the economic reasoning of the variables in the models.
5.1 The Phillips curve
As mentioned the Phillips curve very nicely represents the three pillars of the Key-nesian approach. The ”demand outstripping supply” is represented by the activity variable, the ”built in inflation” is represented by the lagged inflation and also by the constant term, and the supply shock is represented by shock variables. Further-more note that we forecast one-year ahead (four quarters) and two-year ahead (four quarters of only the second year), and we want to take all the available information into account up to that point in time. To do this we need to forecast each quarter in a different manner. Therefore, for the one-year forecast, the model is given by
πt+1= α + β1(L)πt+1+ β2(L)At+1+ β3(L)St+1+ t+1,
πt+2= α + β1(L)πt+1+ β2(L)At+1+ β3(L)St+1+ t+2,
πt+3= α + β1(L)πt+1+ β2(L)At+1+ β3(L)St+1+ t+3,
πt+4= α + β1(L)πt+1+ β2(L)At+1+ β3(L)St+1+ t+4,
and for the two-year forecast, the model is given by
πt+5= α + β1(L)πt+1+ β2(L)At+1+ β3(L)St+1+ t+5,
πt+6= α + β1(L)πt+1+ β2(L)At+1+ β3(L)St+1+ t+6,
πt+7= α + β1(L)πt+1+ β2(L)At+1+ β3(L)St+1+ t+7,
πt+8= α + β1(L)πt+1+ β2(L)At+1+ β3(L)St+1+ t+8,
where π, α, A and S are inflation, a constant, activity variables and shock impulse variables, respectively. The number of lags used for each variable is determined by a model comparison based on the Akaike information criterion.
In the next section, the model selection, it is shown which variable(s) with end up representing the activity variable in the Phillips curve the model.
The variables that are eligible for being a shock variable are the oil prices, the dollar/euro exchange rate, and the world trade. These variables represent a shock to the economy. Only the oil price is a real supply shock. The dollar/euro exchange rate and the world trade can be both supply and demand shocks. For this research the difference is not really important since both have effect on the inflation. In the next section, the model selection, it is shown which variable(s) end up representing the shock variable in Phillips curve the model.
5.1.1 Model selection
The specific model that is selected to represent the Phillips curve in the forecast competition is selected based on an in-sample forecast. We look at the Akaike In-formation criteria (AIC) and the root mean squared error (RMSE) dominantly. The model with the lowest AIC and RMSE is chosen in a systematic way. Since all data for this model are available from 1988Q1-2014Q4, this is the time period that is used to select the best model. The ten most insightful models, in finding the model with the lowest AIC and RMSE, are represented in table 14 in the Appendix.
The table 14 shows that Phillips curve model 10 (inflation regressed on lagged infla-tion, lagged output gab, lagged dollar/euro exchange rate, and lagged world trade) has the lowest RMSE. Phillips curve model 8 (inflation regressed on lagged infla-tion, lagged output gab, and lagged dollar/euro exchange rate) has a slightly higher RMSE and is nested in Phillips curve model 10. Therefore the AIC can decide which model to use. Since the AIC of the Phillips curve model 8 is lower, that model is selected. Furthermore the parameters show a similar pattern as expected from an economic theory.
and the discretionary income of people goes down. The contrary happens in case of a rise in the output gap. Hence there is a positive relationship between inflation and the output gap.
For the dollar/euro exchanges rate a negative relation is found. The economic reason-ing is twofold. First, if your money is worth less abroad, prices are relatively higher, hence inflation. The other reasoning is that there is more demand for Dutch prod-ucts, since these are relatively cheap. This creates more demand, ceterus paribus, resulting in higher prices. Hence a lower dollar/euro exchange rate creates inflation.
Hence the model representing the Phillips curve, for the one-year ahead forecast, is given by
πt+1= α + β1(L)πt+1+ β2(L)OutGapt+1+ β3(L)Dollart+1+ t+1
πt+2= α + β1(L)πt+1+ β2(L)OutGapt+1+ β3(L)Dollart+1+ t+2,
πt+3= α + β1(L)πt+1+ β2(L)OutGapt+1+ β3(L)Dollart+1+ t+3,
πt+4= α + β1(L)πt+1+ β2(L)OutGapt+1+ β3(L)Dollart+1+ t+4,
and for two-years ahead forecast the model is given by
πt+5= α + β1(L)πt+1+ β2(L)OutGapt+1+ β3(L)Dollart+1+ t+5
πt+6= α + β1(L)πt+1+ β2(L)OutGapt+1+ β3(L)Dollart+1+ t+6,
πt+7= α + β1(L)πt+1+ β2(L)OutGapt+1+ β3(L)Dollart+1+ t+7,
πt+8= α + β1(L)πt+1+ β2(L)OutGapt+1+ β3(L)Dollart+1+ t+8,
where π, OutGap and Dollar represent inflation, output gap and the dollar/euro exchange rate. The lags for π, OutGap and Dollar are 3, 1 and 1, respectively. The fact that three lags of inflation return in the Phillips curve could be explained by the persistence of inflation.
5.2 New Keynesian Phillips curve
year), and we want to take all the available information into account up to that point in time. To do this we need to forecast each quarter in a different manner. Therefore, for the one-year forecast, the model is given by
πt+1 = α + β1(L)πt+1+ β2(L)At+1+ β3(L)St+1+ β4Ft−3+ t+1
πt+2 = α + β1(L)πt+1+ β2(L)At+1+ β3(L)St+1+ β4Ft−2+ t+2,
πt+3 = α + β1(L)πt+1+ β2(L)At+1+ β3(L)St+1+ β4Ft−1+ t+3,
πt+4 = α + β1(L)πt+1+ β2(L)At+1+ β3(L)St+1+ β4Ft+ t+4,
and for the two-year forecast, the model is given by
πt+5 = α + β1(L)πt+1+ β2(L)At+1+ β3(L)St+1+ β4Ft−3+ t+5
πt+6 = α + β1(L)πt+1+ β2(L)At+1+ β3(L)St+1+ β4Ft−2+ t+6,
πt+7 = α + β1(L)πt+1+ β2(L)At+1+ β3(L)St+1+ β4Ft−1+ t+7,
πt+8 = α + β1(L)πt+1+ β2(L)At+1+ β3(L)St+1+ β4Ft+ t+8,
where π, α, A, S, and F are inflation, a constant, activity variables, shock impulse variables, and forward looking variables, respectively. The number of lags used for each variable is determined by a model comparison based on the Akaike information criterion.
Note that the time of the forward looking variable is unusual, this is because the forward looking variable looks one year into the future, which is 4 quarters. For the two-year forecast the forward looking variable is the expected value of of last year on the based on the knowledge 2 years ago. So it is looking less far into the future than the forecast itself. While for the one-year ahead forecasts these horizons are equal.
prove, deteriorate or remain the same in the next 12 months?” and the Financial Consumer Confidence of the coming 12 months answers the question: ”Will the fi-nancial situation of your household improve, deteriorate or remain the same in the next 12 months?”. The output is the balance of the positive and negative feed-back as a percentage. In the Model selection it is shown which variable(s) end up representing the forward looking variable in the New Keynesian Phillips curve. 5.2.1 Model selection
The specific model that is selected to represent the New Keynesian Phillip curve in the forecast competition is selected based on an in-sample forecasting in a similar way as by the Phillips curve. The difference here is that we do need at least one variable of the forward looking variables, otherwise the model could be similar to the Keynesian model. Since all data for this model are available from 1999Q1-2014Q4, that sample is used to select the best model. The five most insightful models, are represented in table 15 in the Appendix.
In table 15 we see that New Keynesian Phillips curve model 2 (inflation regressed on lagged inflation, lagged output gab, lagged dollar/euro exchange rate, and the expected inflation of professional) has the lowest RMSE and has the lowest AIC, therefore that model is selected. Furthermore the parameters show a similar pattern as expected from economic theory.
Hence the model that represents the New Keynesian Phillips curve, for the one-year ahead forecast, is given by
πt+1= α + β1(L)πt+1+ β2(L)OutGapt+1+ β3(L)Dollart+1+ Et−3(πt+1) + t+1
πt+2= α + β1(L)πt+1+ β2(L)OutGapt+1+ β3(L)Dollart+1+ Et−2(πt+2) + t+2,
πt+3= α + β1(L)πt+1+ β2(L)OutGapt+1+ β3(L)Dollart+1+ Et−1(πt+3) + t+3,
and for two-years ahead forecast the model is given by
πt+5= α + β1(L)πt+1+ β2(L)OutGapt+1+ β3(L)Dollart+1+ Et−3(πt+1) + t+5
πt+6= α + β1(L)πt+1+ β2(L)OutGapt+1+ β3(L)Dollart+1+ Et−2(πt+2) + t+6,
πt+7= α + β1(L)πt+1+ β2(L)OutGapt+1+ β3(L)Dollart+1+ Et−1(πt+3) + t+7,
πt+8= α + β1(L)πt+1+ β2(L)OutGapt+1+ β3(L)Dollart+1+ Et(πt+4) + t+8,
where π, α, OutGap, Dollar, Et(πt+4) represent inflation, a constant output gap,
the dollar/euro exchange rate and expected inflation, respectively.
The lags for π, OutGap, and Dollar are 3, 1, and 1, respectively.
5.3 Monetary/Financial Phillips curve
Monetary/Financial Phillips curve relates to the Taylor rule, and therefore also re-lates to the New Keynesian view. This model will, similar to NKPC, be inserted from the Phillips curve only now by adding monetary and financial variables. Fur-thermore note that, as with the Phillips Curve and NKPC, we forecast one-year ahead (four quarters) and two-year ahead (four quarters of only the second year), and we want to take all the available information into account up to that point in time. To do this we need to forecast each quarter in a different manner. Therefore, for the one-year forecast, the model is given by
πt+1= α + β1(L)πt+1+ β2(L)At+1+ β3(L)St+1+ β4(L)Mt+1+ t+1
πt+2= α + β1(L)πt+1+ β2(L)At+1+ β3(L)St+1+ β4(L)Mt+1+ t+2,
πt+3= α + β1(L)πt+1+ β2(L)At+1+ β3(L)St+1+ β4(L)Mt+1+ t+3,
πt+4= α + β1(L)πt+1+ β2(L)At+1+ β3(L)St+1+ β4(L)Mt+1+ t+4,
and for the two-year forecast, the model is given by
πt+5= α + β1(L)πt+1+ β2(L)At+1+ β3(L)St+1+ β4(L)Mt+1+ t+5
πt+6= α + β1(L)πt+1+ β2(L)At+1+ β3(L)St+1+ β4(L)Mt+1+ t+6,
πt+7= α + β1(L)πt+1+ β2(L)At+1+ β3(L)St+1+ β4(L)Mt+1+ t+7,
variables, and monetary and financial variables, respectively. The number of lags used for each variable is determined by a model comparison based on the Akaike information criterion.
The variables that are eligible to represent the Monetary and Financial variables are the marginal lending facility, the Amsterdam Exchange Index (AEX), and the Gold price in dollars. The marginal lending facility is determined by the ECB after 2000 and by the marginal lending facility of the DNB before 2000. The marginal lending facility is a monetary variable, this is also used in the Taylor rule. The AEX and the Gold price are financial variables. In the Model selection it is shown which variable(s) end up representing the monetary and financial variable(s) in the Monetary/Financial Phillips curve.
5.3.1 Model selection
The specific model that is selected to represent the Monetary and Financial approach in the forecast competition is selected based on an in-sample forecasting in a way similar to the Phillips curve and the New Keynesian Phillips curve approach. The difference here is that we do need at least one variable of the Monetary and Financial variables, otherwise the model could be similar to the Keynesian model. Since all data for this model is available from 1988Q1-2014Q4, this is the time period that is used to select the best model. The most insight full models are represented in the table 16 in the Appendix.
In table 16 we can see that Model 5 (inflation regressed on lagged inflation, lagged output gab, lagged dollar/euro exchange rate, lagged first difference of AEX, and lagged first difference the gold price) has both the lowest RMSE as well as the lowest AIC, therefore this model is be selected. Furthermore the parameters show a similar pattern as expected from economic theory.
for the one-year ahead forecast, is given by πt+1= α + β1(L)πt+1+ β2(L)OutGapt+1+ β3(L)Dollart+1 + β4D.AEXt+1+ β5(L)D.Goldt+1+ t+1 πt+2= α + β1(L)πt+1+ β2(L)OutGapt+1+ β3(L)Dollart+1 + β4D.AEXt+1+ β5(L)D.Goldt+1+ t+2, πt+3= α + β1(L)πt+1+ β2(L)OutGapt+1+ β3(L)Dollart+1 + β4D.AEXt+1+ β5(L)D.Goldt+1+ t+3, πt+4= α + β1(L)πt+1+ β2(L)OutGapt+1+ β3(L)Dollart+1 + β4D.AEXt+1+ β5(L)D.Goldt+1+ t+4,
and for two-years ahead forecast the model is given by
πt+5= α + β1(L)πt+1+ β2(L)OutGapt+1+ β3(L)Dollart+1 + β4D.AEXt+1+ β5(L)D.Goldt+1+ t+5 πt+6= α + β1(L)πt+1+ β2(L)OutGapt+1+ β3(L)Dollart+1 + β4D.AEXt+1+ β5(L)D.Goldt+1+ t+6, πt+7= α + β1(L)πt+1+ β2(L)OutGapt+1+ β3(L)Dollart+1 + β4D.AEXt+1+ β5(L)D.Goldt+1+ t+7, πt+8= α + β1(L)πt+1+ β2(L)OutGapt+1+ β3(L)Dollart+1 + β4D.AEXt+1+ β5(L)D.Goldt+1+ t+8,
where π, α, OutGap, Dollar, D.AEX and D.Gold represent inflation, a constant, output gap, the dollar/euro exchange rate, the first difference of the AEX and the first difference of the Gold price in dollars.
The lag variables are determined based on the Akaike information criterion (AIC). The lags for π, OutGap, Dollar, D.AEX and D.Gold are 3, 1, 1, 1, and 1, respec-tively.
5.4 Bayesian VAR
Y = XB + U
y = Im⊗ Xβ + u u ∼ 0, Σu⊗ IT)
where Y and U are T × m matrices and X is a T × k matrix. Furthermore Xt = [ιT, yt−1T , . . . , yt−qT , ¯yTt]T, where ι is a vector of ones, and y and u are mT × 1
vectors, Im is the identity matrix of dimension m and β = vec(B) is a mk × 1 vector.
The likelihood function is L β, Σu ∝ Σu⊗ IT −0.5 exp − 0.5y − (Im⊗ X)β T Σ−1u ⊗ IT y − (Im⊗ X)β
Let the prior be β = ¯β + vβ, vβ ∼ N (0, ¯Σβ), where ¯β and ¯Σβ are a function of
a small number of hyperparameters. We assume most of all that ¯βi1 = 0.15, for
i = 1, . . . , m and zero otherwise. Furthermore we assume that ¯Σβ is diagonal and
that σij,l element is corresponding to lag l of variable j in equation i has the form:
σij,l= φ0 h(l) if i = j, ∀l = φ0 φ1 h(l) σj σi 2
otherwise when i 6= j, j endogenous, ∀l
= φ0φ2 for j exogenous
Where φi, for i = 0, 1, 2, are hyperparameter, (σσji)2 is a scaling factor and h(l) a
deterministic function of l. This is called the Minnesota prior. Therefore the Min-nesota prior variance is characterized by the tightness on variance of the first lag (φ0), the relative tightness of other variables (φ1) and the relative tightness of the
exogenous variables (φ2) and the relative tightness of the variance of lags other than
the first one (h(l)). Since σi, i = 1, . . . , m are unknown, consistent estimates of the
standard errors of the variables i, j are used.
The prior is a-priori a stationary walk and updates with the information represented by the data, this is particularly good in forecasting macroeconomic stationary time series. Therefore we have chosen this prior, which is developed from a Minnesota prior. Note that we can optimally chose φi, for i = 0, 1, 2. For the analysis we set
6
Empirical results
The data used for this forecast competition starts at the same date 1999Q1, to make sure that all models are comparable in a fairly fashion. The forecast competition is done over to time lines a one-year ahead forecast and a two-year ahead forecast. So for example for the one-year ahead forecast of 2008, we will use the sample until 2007. For the two-year ahead forecast of 2008; the sample is used until 2006. After providing the result of the one-year ahead forecast of both the quarterly and yearly forecasts, the same is provided for the two-year ahead forecasts.
6.1 One-year ahead forecasts
In this section the results are displayed of the one-year ahead forecast reported on a quarterly basis and a yearly basis. Only the forecast of SAFFIER model are available on a yearly basis, therefore this model is not mentioned in the quarterly forecast. Furthermore we test if the models are significant outperforming each other. Also we will look if the models systematically over or underestimating the inflation.
6.1.1 Forecasts on quarterly basis
In Figure 3 the forecasts are displayed for the quarterly data of the one-year ahead forecast; also the actual inflation is shown in this figure.
In the table 1, the mean absolute forecast error (MAFE) and the root mean squared forecast error (RMSFE) of the forecasts are displayed.
Statistic Phillips curve (PC) NKPC MFPC BVAR MAFE 0.2735 0.2492 0.2784 0.2050 RMSFE 0.3528 0.3225 0.3567 0.2979 Table 1: Statistics of one-year ahead forecasts on quarterly basis
6.1.2 Forecasts on yearly basis
In Figure 4 the yearly forecasts are displayed; also the actual inflation is shown in this figure.
Figure 4: One-year ahead forecast on yearly basis In the table ??, the MAFE and RMSFE of the forecasts are displayed.
6.1.3 Significance of the difference: bootstrap
In the bootstrap we created 1000 sets of forecast errors from the original set of fore-cast errors. By the law of large numbers the models can be described as a normal distribution with mean and standard deviation. The mean and standard deviation are obtained from the bootstrap. In table 2 and table 3 the results of the bootstrap based on the quarterly forecast errors and the yearly forecast error are shown.
Statistic PC NKPC MFPC BVAR MAFE Mean 0.2746 0.2439 0.2755 0.2090 MAFE Standard deviation 0.0401 0.0338 0.087 0.0371 RMSFE Mean 0.3480 0.3094 0.3470 0.2942 RMSFE Standard deviation 0.0448 0.0402 0.0433 0.0508
Table 2: Mean & Standard deviation obtained from the bootstrap for one-year ahead forecasts on quarterly basis
Statistic SAFFIER PC NKPC MFPC BVAR MAFE Mean 0.2123 0.7106 0.6204 0.7437 0.3067 MAFE Standard deviation 0.0605 0.0153 0.1259 0.1416 0.1006 RMSFE Mean 0.2568 0.8364 0.7309 0.8214 0.3810 RMSFE Standard deviation 0.0624 0.1358 0.1023 0.1477 0.1229 Table 3: Mean & Standard deviation obtained from the bootstrap for one-year ahead forecasts on yearly basis
6.1.4 Significance of the difference: Tests
We test if one model outperforms the other model. This is done by P(MAFEx< MAFEy) ≤ 0.05,
equivalently
P(MAFEx− MAFEy < 0) ≤ 0.05,
or
P(D < 0) ≤ 0.05,
where D = MAFEx− MAFEy and subscript x and y represent different models. By
given by
E(D) = E(MAFEx− MAFEy),
= E(MAFEx) − E(MAFEy),
Var(D) = VAR(MAFEx− MAFEy),
= Var(MAFEx) + Var(MAFEy) − 2Cov(MAFEx, MAFEy).
The t-statistic of the test is given by
t = E(D) pVar(D)
If the t ≤ −1.645 we cannot reject the null hypotheses that model Y outperforms model X.
The SAFFIER, the BVAR, and the Phillip curves are constructed in very differ-ent ways, therefore we assume their covariance is zero. This also ensures that we do not falsely reject the hypothesis. For the different Phillip curves we cannot assume this. The differences in the mean are so small, that the t-statistic will never be significant even if the covariance is not zero. The test is done based on the MAFE, since this statistic gives the clearest relationship with the underlying data.
In table 4 the mean, variance and t-statistic and the different models to be com-pared are displayed for the quarterly basis forecasts.
Statistic BVAR vs PC BVAR vs NKPC BVAR vs MFPC Mean -0.0656 -0.0349 -0.0666 Variance 0.0030 0.0025 0.0029 T-statistic -1.2016 -0.6963 -1.2429
Table 4: Significance tests based on one-year ahead forecasts on quarterly basis All the t-values in table 4 show the same result. We cannot reject the hypothesis that one model is as good a predictive model as than the other model, since the since the t-value are between -1.645 and 1.645.
Statistic SAFFIER vs NKPC SAFFIER vs BVAR BVAR vs NKPC
Mean -0.4081 -0.0944 -0.3138
Variance 0.0272 0.0138 0.0336
T-statistic -2.4747 -0.8041 -1.7104 Table 5: Significance tests based on one-year ahead forecasts on yearly basis The t-statistics in table 5 show that we can reject the hypotheses that the benchmark SAFFIER and PC’s have equal forecast power and that the BVAR and the PC’s have equal forecast power both in favour of the BVAR since -2.4747 and -1.7104 are both smaller than -1.645. Furthermore the table shows us that we cannot reject the null hypothesis that BVAR is as good a predictive model as the benchmark SAFFIER, since the t-value is −1.645 < −0.8041 < 1.645.
6.1.5 Systematic error
The reason that the BVAR does not significantly outperforms the Phillips curves in the yearly forecast, but not in quarterly forecast could be due a higher systematic error. By creating a bigger forecast period the errors can cancel each other out, the bigger the systematic error the higher the chance that they do not cancel each other out. So it is possible to systematic overestimate inflation or underestimate inflation. To check if this is indeed what happened we calculate the systematic error of the quarterly forecasts by the mean forecast error, which is given by
MFE = 1 n n X t=1 Ft− At,
where At is the actual level of inflation at time t, Ft is the forecast level of inflation
at time t and n is the number of forecasts in the sample.
In table 6, the mean forecast errors of all the models are shown. Statistic PC NKPC MFPC BVAR
MFE -0.0481 -0.0935 -0.0131 0.0153
6.2 Two-year ahead forecasts
In this section the results are displayed of the two-year ahead forecast reported on a quarterly basis and a yearly basis. Only the forecasts of SAFFIER model are available on a yearly basis, therefore this model is not mentioned in the quarterly forecast. Furthermore we test if the models are significant outperforming each other. 6.2.1 Forecasts on quarterly basis
In Figure 5 the forecasts are displayed for the quarterly data of the two-year ahead forecast; also the actual inflation is shown in this figure.
Figure 5: Two-year ahead forecast on quarterly basis In the table 7, the MAFE and the RMSFE of the forecasts are displayed.
Statistic PC NKPC MFPC BVAR MAFE 0.4162 0.4560 0.4516 0.3077 RMSFE 0.6077 0.6365 0.5905 0.3784
6.2.2 Forecasts on yearly basis
In Figure 6 the forecasts are the yearly displayed; also the actual inflation is shown in this figure.
Figure 6: Two-year ahead forecast on yearly basis In the table 8, the MAFE and RMSFE of the forecasts are displayed.
Statistic SAFFIER PC NKPC MFCPC BVAR MAFE 0.8723 1.5720 1.7454 1.6568 0.8018 RMSFE 0.9834 2.0046 2.1367 1.6651 0.9007 Table 8: Statistics of two-year ahead forecasts on yearly basis
6.2.3 Significance of the difference: bootstrap & tests
In table 9 and table 10 the results of the bootstrap based on the quarterly forecasts errors and the yearly forecasts error are displayed.
Statistic PC NKPC MFPC BVAR MAFE Mean 0.4141 0.4568 0.4528 0.3066 MAFE Standard deviation 0.0795 0.0760 0.06737 0.0381 RMSFE Mean 0.6015 0.3392 0.5828 0.3761 RMSFE Standard deviation 0.1202 0.0346 0.0879 0.0451
Table 9: Mean & Standard deviation obtained from the bootstrap of two-year ahead forecasts on quarterly basis
Statistic SAFFIER PC NKPC MFPC BVAR MAFE Mean 0.8682 1.5683 1.7275 1.6479 0.8042 MAFE Standard deviation 0.1752 0.4790 0.4720 0.3272 0.1489 RMSFE Mean 0.9662 1.9206 2.059 1.8436 0.8803 RMSFE Standard deviation 0.1634 0.5543 0.5187 0.3785 0.1759 Table 10: Mean & Standard deviation obtained from the bootstrap of two-year ahead forecasts on yearly basis
As for the one-year ahead forecasts, also the two-year ahead forecasts are tested if one model significantly outperforms the other models. In table 11 the mean, variance and t-statistic and the different models to be compared are displayed for the quarterly basis forecasts provided by a bootstrap.
Statistic BVAR vs PC BVAR vs NKPC BVAR vs MFPC Mean -0.1075 -0.1502 -0.1462 Variance 0.0078 0.0072 0.0060 T-statistic -1.2197 -1.7667 -1.8887
In table 12 the mean, variance and t-statistic and the different models to be com-pared are displayed for the yearly data basis forecasts.
Statistic SAFFIER vs NKPC SAFFIER vs BVAR BVAR vs NKPC
Mean -0.8593 0.0639 -0.9232
Variance 0.2535 0.0529 0.2450
T-statistic -1.7067 0.2779 -1.8653
7
Conclusion
In this study we investigate the forecasting power of different models relating to different economic views. These models will be benchmarked against SAFFIER. Forecasts are made one-year ahead and two-year ahead. Furthermore the forecasts are reported based on quarterly basis and on yearly basis.
In the one-year ahead forecast and two-year ahead forecast on yearly basis we find that the BVAR significantly outperforms the different versions of the Phillips curve, but is does not significantly outperform the benchmark model SAFFIER.
In the one-year ahead forecast and two-year ahead forecast on quarterly basis we find that the BVAR does significantly outperforms the different versions of the Phillips curve for the one-year ahead and for the Phillips curve of two-year ahead forecast. For the New Keynesian Phillips curve and the Monetary/Financial Phillips curve for find that the BVAR is significantly better at forecasting inflation on the two-year horizon. Furthermore we find that the BVAR and SAFFIER are statistically similar for the one-year ahead forecast and two-year ahead forecast on quarterly bases.
The reason that the BVAR does not significantly outperform the different version of the Phillips curves on a quarterly basis, but it does in on a yearly basis for the one-year ahead forecast, can not be explained due to a difference in systematic error term of the forecasts.
Also, the result hints to the conclusion that the Phillips curves do not significantly outperform each other. It could be that the gab between the initial models and the-oretical background has become smaller as the Phillip curves developed over time.
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Variable Model 1 Model 2 Model 3 Model 4 Model 5 α 1.172349 .7102922 1.313974 1.376009 .8588822 (2.50) (1.56) (2.69) (1.90) (1.78) L1.π .0415345 .0251993 .0180011 .0255157 -.0002898 (0.33) (0.20) (0.14) (0.21) (-0.00) L2.π -.0063987 .0046391 -.0427607 -.0322956 (-0.07) (0.06) (-0.45) (0.27) (-0.38) L3.π .0829598 .1187956 .0322648 .0550258 .0644964 (0.62) (0.87) (0.20) (0.35) (0.40) L1.OutGap .0961711 .0810714 .104322 .1099399 .0897217 (3.33) (2.58) (3.48) (2.82) (2.83) L1.Dollar -.5969794 -.5427619 -.6774496 -.7400939 -.628976 (-1.88) (-1.84) (-2.09) (-1.47) (-2.09) Et(πt+1) .2246345 .2265608 (1.66) (1.66) ConConEco -.0011882 -.0012799 -0.61 (-0.72) ConConF in -.0029712 (-0.41) RMSE .28844 .28416 .29038 .2905 .28598 R2 0.3680 0.3978 0.3711 0.3706 0.4013 F-statistic 5.02 4.93 4.27 4.08 4.71 AIC 27.11 26.17 28.81 28.86 27.81 Table 15: Model Selection New Keynesian Phillips curve
Variable Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 α .7995069 .7174842 .7381477 .8273529 .7637976 .6925366 (2.60) (2.15) (2.47) (2.64) (2.55) (2.11) L1.π .1129073 .0990405 .1174634 .090363 .0906011 .0779377 (1.36) (1.18) (1.47) (1.04) (1.07) (0.91) L2.π .089233 .0764046 .1048526 .1236035 .1488548 .1367344 (0.98) (0.90) (1.17) (1.26) (1.52) (1.47) L3.π .1411709 .1304556 .1754242 .1481682 .1885804 .1775961 (1.55) (1.41) (2.04) (1.59) (2.15) (1.97) L1.OutGap .0789095 .0712211 .0862754 .0760945 .0838395 .0766346 (3.34) (2.77) (3.81) (3.15) (3.66) (3.07) L1.Dollar -.3663069 -.3318723 -.3426405 -.4021937 -.3831832 -.3532042 (-1.61 ) (-1.39) (-1.56) (-1.68) (-1.68) (-1.48) L1.MargLen .0144606 .0129969 (0.94) (0.86) L1.D.AEX .0012478 .0014164 .0013668 (1.76) (1.93) (1.87) L1.D.Gold .0007 .0008532 .0008538 (1.38) (1.66) (1.66) RMSE .27505 .27501 .27295 .27444 .27139 .27160 R2 0.3127 0.3198 0.3299 0.3226 0.3444 0.3501 F-statistic 6.71 5.95 7.34 5.96 6.50 5.77 AIC 32.728 33.63 32.06 33.20 31.77 32.85 Table 16: Model Selection Monetary/Financial Phillips curve
